Recently Active Olympiad Stuff Questions

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prove that there exists a triangle which can be cut into 2005 congruent triangles ...

1) define N=2n(2n+1 - 1) , nEN... such dat (2n+1-1) is a prime no. prove that:- i) the sum of the divisors of N are 2N. ii) the sum of the reciprocals of the divisors of N are 2. 2) find all continuous functions f : (0, infi) ...

In the triangle ABC , B and C are acute angles. The altitude of the triangle drawn from Aintersects BC at D . The bisectors of the angles B and C intersect AD at E and F respectively . If BE = CF , Prove that the triangle ABC ...

find the no of positive integral solutions of \left[ \frac{x}{900}\right]=\left[ \frac{x}{901}\right] ...

Five pirates and a monkey are shipwrecked on an island. The pirates have collected a pile of coconuts which they plan to divide equally among themselves the next morning. Not trusting the others, one pirate wakes up during th ...

Q A=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+................+\frac{1}{\sqrt{10000}} Then find [A] where [] represents GINT ...

Found somewhere.. very good problem, check how trivially you can give answer..... Is there an infinite matrix Amn such that limn→∞ amn=0 for every m and limm→∞ amn=1 for every n ? ...

ok lets have sum fun in number thoery with some questions. here is first one 1] let x and y be positive integers such that xy divides x^{2}+y^{2}+1 . show that \frac{x^{2}+y^{2}+1}{xy}=3 ...

Find the number of quadratic polynomials ax^2+bx+c which satisfy the following simultaneously: (a) a,b,c are distinct; (b) a,b,c\in \left\{ 1,2,3,4.............. 2009\right\} ; (c) (x+1) divides ax^2+bx+c ...

Find all positive integral solutions (a,b,c) to the equation (a+1)(b+1)(c+1)=2(abc + 1) ...

The product of the digits of a three-digit number is 105. What is the largest possible value of the number? its a sitter....just about 2 lines.. ...

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try to solve it in a elegant way..... A circular table is divided into five sectors. You can paint each sector with one of five colors. If more than one sector can be painted the same color, how many ways are there to paint t ...

n=2p.3q how many factors of n2 are smaller than n (edit) but do not divide n? ...

At vertices of a convex polygon with 2n sides, sit hunters. while in its interior and not on any diagonal,sits a fox. The hunters shoot at the fox all simultaneously, but the fox ducks. the bullets hit the sides of the polygo ...

Prove that any finite set of closed squares with total area 3 can be arranged to cover the unit square. ...

sveral numbers are written in a row. in each move, john choses any 2 adjacent numbers in which one on the left is greater than one on the right, doubles each of them and switches around. prove that roberts cn make only a fini ...

let f be continuous for any x>1 and [] denoting the greatest integer function prove that \int_{1}^{x}(+1)f(u)du =2\sum_{i=1}^{[x]}{i\int_{i}^{x}{f(u)du}} ...

a good user was posting some good questions..........so here is one from my side.....very easy...[3][3] let f be twice diff real valued function satisfying f(x)+f"(x)=-x.g(x).f'(x) wheere g(x)≥0 for all real x. Prove that m ...

All points of the plane are colored red, blue or green. Prove that we can find two points a distance 1 apart with the same color. ...

Let m, n, p, q, r, s be positive integers such that p < r < m and q < s < n. In how many ways can one travel on a rectangular grid from (0, 0) to (m, n) such that at each step one of the coordinates increases by o ...

*Image* AND NO PRIZES FOR GUESSING THE SOURCE :p ...

Q Solve for x x+a^{3}=\sqrt[3]{a-x } ...

A device consists of 4n elements, any two of which are joined by either a red or a blue wire. The numbers of red and blue wires are the same. The device is disabled by removing two wires of the same color connecting four di e ...

Q prove the following inequality \sqrt[3]{3-\sqrt[3]{3}}+\sqrt[3]{3+\sqrt[3]{3}}<2\sqrt[3]{3} ...

Find all positive integral solutions (a,b,c) to the equation (a+1)(b+1)(c+1)=2(abc + 1) ...

The equation ax3 +bx2 +cx+d = 0 is known to have three distinct real roots. How many real roots are there of the equation 4(ax3 + bx2+ cx + d)(3ax + b) = (3ax2 + 2bx + c)2? ...

let N be positive integer with 1998 decimal digits all of them 1 such that n=1111.......11 Find the thousandth digit after decimall point of N ...

Twenty-one girls and twenty-one boys took part in a mathematics competition. It turned out that (i) each contestant solved at most six problems, and (ii) for each pair of a girl and a boy, there was at least one problem that ...